double pdtr( int k, double m ) { double v; if( (k < 0) || (m <= 0.0) ) { mtherr( "pdtr", DOMAIN ); return( 0.0 ); } v = k+1; return( igamc( v, m ) ); }
static DBL igam(DBL a, DBL x) { DBL ans, ax, c, r; int sgngam = 0; if ((x <= 0) || (a <= 0)) { return (0.0); } if ((x > 1.0) && (x > a)) { return (1.0 - igamc(a, x)); } /* Compute x**a * exp(-x) / gamma(a) */ ax = a * log(x) - x - lgam(a, &sgngam); if (ax < -MAXLOG) { /* mtherr("igam", UNDERFLOW); */ return (0.0); } ax = exp(ax); /* power series */ r = a; c = 1.0; ans = 1.0; do { r += 1.0; c *= x / r; ans += c; } while (c / ans > MACHEP); return (ans * ax / a); }
double igamci(double a, double q) { int i; double x, fac, f_fp, fpp_fp; if (npy_isnan(a) || npy_isnan(q)) { return NPY_NAN; } else if ((a < 0.0) || (q < 0.0) || (q > 1.0)) { mtherr("gammainccinv", DOMAIN); } else if (q == 0.0) { return NPY_INFINITY; } else if (q == 1.0) { return 0.0; } else if (q > 0.9) { return igami(a, 1 - q); } x = find_inverse_gamma(a, 1 - q, q); for (i = 0; i < 3; i++) { fac = igam_fac(a, x); if (fac == 0.0) { return x; } f_fp = (igamc(a, x) - q) * x / (-fac); fpp_fp = -1.0 + (a - 1) / x; if (npy_isinf(fpp_fp)) { x = x - f_fp; } else { x = x - f_fp / (1.0 - 0.5 * f_fp * fpp_fp); } } return x; }
double igami( double a, double y0 ) { double x0, x1, x, yl, yh, y, d, lgm, dithresh; int i, dir; /* bound the solution */ x0 = MAXNUM; yl = 0; x1 = 0; yh = 1.0; dithresh = 5.0 * MACHEP; /* approximation to inverse function */ d = 1.0/(9.0*a); y = ( 1.0 - d - ndtri(y0) * sqrt(d) ); x = a * y * y * y; lgm = lgam(a); for( i=0; i<10; i++ ) { if( x > x0 || x < x1 ) goto ihalve; y = igamc(a,x); if( y < yl || y > yh ) goto ihalve; if( y < y0 ) { x0 = x; yl = y; } else { x1 = x; yh = y; } /* compute the derivative of the function at this point */ d = (a - 1.0) * log(x) - x - lgm; if( d < -MAXLOG ) goto ihalve; d = -exp(d); /* compute the step to the next approximation of x */ d = (y - y0)/d; if( fabs(d/x) < MACHEP ) goto done; x = x - d; } /* Resort to interval halving if Newton iteration did not converge. */ ihalve: d = 0.0625; if( x0 == MAXNUM ) { if( x <= 0.0 ) x = 1.0; while( x0 == MAXNUM ) { x = (1.0 + d) * x; y = igamc( a, x ); if( y < y0 ) { x0 = x; yl = y; break; } d = d + d; } } d = 0.5; dir = 0; for( i=0; i<400; i++ ) { x = x1 + d * (x0 - x1); y = igamc( a, x ); lgm = (x0 - x1)/(x1 + x0); if( fabs(lgm) < dithresh ) break; lgm = (y - y0)/y0; if( fabs(lgm) < dithresh ) break; if( x <= 0.0 ) break; if( y >= y0 ) { x1 = x; yh = y; if( dir < 0 ) { dir = 0; d = 0.5; } else if( dir > 1 ) d = 0.5 * d + 0.5; else d = (y0 - yl)/(yh - yl); dir += 1; } else { x0 = x; yl = y; if( dir > 0 ) { dir = 0; d = 0.5; } else if( dir < -1 ) d = 0.5 * d; else d = (y0 - yl)/(yh - yl); dir -= 1; } } if( x == 0.0 ) mtherr( "igami", UNDERFLOW ); done: return( x ); }
inline double gamma_q(double x, double y) { return igamc(x, y); }
static DBL igami(DBL a, DBL y0) { DBL d, y, x0, lgm; int i; int sgngam = 0; /* approximation to inverse function */ d = 1.0 / (9.0 * a); y = (1.0 - d - ndtri(y0) * sqrt(d)); x0 = a * y * y * y; lgm = lgam(a, &sgngam); for (i = 0; i < 10; i++) { if (x0 <= 0.0) { /* mtherr("igami", UNDERFLOW); */ return (0.0); } y = igamc(a, x0); /* compute the derivative of the function at this point */ d = (a - 1.0) * log(x0) - x0 - lgm; if (d < -MAXLOG) { /* mtherr("igami", UNDERFLOW); */ goto done; } d = -exp(d); /* compute the step to the next approximation of x */ if (d == 0.0) { goto done; } d = (y - y0) / d; x0 = x0 - d; if (i < 3) { continue; } if (fabs(d / x0) < 2.0 * MACHEP) { goto done; } } done: return (x0); }